On commuting automorphisms of finite p-groups of co-class 3, when $p>2$

Let G be a group. An automorphism α of G is called a commuting automorphism if [α(g),g]=1 for all g ∈ G. Let $A(G)$ denote the set of all commuting automorphisms of G. A group G is said to be an $A(G)$-group if $A(G)$ forms a subgroup of Aut(G), where Aut(G) denotes the group of all automorphisms of G. In [Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 5, 57-60] Rai proved that a finite p-group G of co-class 2 for an odd prime p is an $A(G)$-group. We prove that a finite p-group G of co-class 3 for an odd prime p, under some conditions, is an $A(G)$-group.